Global well-posedness and blow-up for the 2-D Patlak–Keller–Segel equation

In this paper, we prove that for every nonnegative initial data in $L^1({\mathbb{R}}^2)$, the Patlak–Keller–Segel equation is globally well-posed if and only if the total mass $M\le 8\pi$. Our proof is based on some monotonicity formulas of nonnegative mild solutions.     

Dunkl-Schrödinger semigroups and applications

We obtain dispersive estimates for the linear Dunkl–Schrödinger equations with and without quadratic potential. As a consequence, we prove the local well-posedness for semilinear Dunkl–Schrödinger equations with polynomial nonlinearity in certain magnetic field. Furthermore, we study many applications: as the uncertainty principles for the Dunkl transform via the Dunkl–Schrödinger semigroups, the embedding theorems for the Sobolev spaces associated with the generalized Hermite semigroup. Finally, almost every where convergence of the solutions of the Dunkl–Schrödinger equation is also considered.     

Antihypertensive drugs as an inhibitors for corrosion of aluminum and aluminum silicon alloys in aqueous solutions

The corrosion behavior of aluminum and three aluminum–silicon alloys in different concentrations of HCl solutions and its inhibition by antihypertensive drugs was studied using potentiostatic polarization measurements. As the acid concentration increases, the rate of corrosion increases. Aluminum is less susceptible to corrosion than any of Al–Si alloys. The inhibition efficiency of the drug compounds increases with their concentration up to a critical value. At higher additive concentrations the inhibition efficiency starts to decrease. The inhibitive action of these compounds is due to their formation of insoluble complex adsorbed on the metal surface. The adsorption follows Langmuir adsorption isotherms. It was found that the drugs compounds provide protection to Al and Al–Si alloys against pitting corrosion by shifting the pitting potential to more positive direction until critical drug concentrations (250 ppm). After this critical concentration the inhibition against to pitting corrosion starts to decrease.     

Diversified Excited-State Relaxation Pathways of Donor–Linker–Acceptor Dyads Controlled by a Bent-to-Planar Motion of the Donor

Herein, we introduce the cyclic 8π-electron (C8π) molecule N,N′-diaryl-dihydrodibenzo[a,c]phenazine ( DPAC ) as a dual-functional donor to establish a series of new donor–linker–acceptor (D–L–A) dyads DLA1 – DLA5 . The excited-state bent-to-planar dynamics of DPAC regulate the energy gap of the donor, while the acceptors A1 – A5 are endowed with different energy gaps and HOMO/LUMO levels. As a result, the rate and efficiency of the excited-state electron transfer vs. energy transfer can be finely harnessed, which is verified via steady-state spectroscopy and time-resolved emission measurements. This comprehensive approach demonstrates, for the first time, the manifold of excited-state properties governed by bifunctional donor-based D–L–A dyads, including bent-to-planar, photoinduced electron transfer (PET) from excited donor to acceptor (oxidative-PET), fluorescence resonance energy transfer (FRET), bent-to-planar followed by electron transfer (PFET), and PET from donor to excited acceptor (reductive-PET).     

Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients

It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada–Watanabe theorem (Yamada and Watanabe, 1971, [31,32]) and the Feller test for explosions (Feller, 1951, 1954), there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain $D$ of the phase space $\mathbb{R}$ and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called free energy function on the density function space, we prove that the density functions, which are solutions of the Fokker–Planck equation, converge to the stationary density function exponentially under the Kullback–Leibler divergence, thus also in the total variation norm. The results show that there is a relation between the Bakry–Émery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher–Lamperti transformation. Several applications are discussed, including the Cox–Ingersoll–Ross model and the Ait-Sahalia model in finance and the Wright–Fisher model in population genetics.     

Spin and charge Nernst effect in a four-terminal double-dot interferometer

We investigate the spin and charge Nernst effect of a four-terminal Aharonov–Bohm interferometer with Rashba spin–orbit interaction (RSOI). It is shown that a pure spin Nernst effect or a fully spin-polarized Nernst effect can be obtained by modulating the magnetic flux phase ? and the RSOI induced phase φ. It is also demonstrated that some windows of ? (or φ) for maintaining an almost fully spin-polarized Nernst effect exist and their width is under the control of the other phase. Moreover, for the charge Nernst coefficient $N_c$ and spin Nernst coefficient $N_s$ the relationship $N_c(?,\phi )=?N_s(\phi ,?)$ always holds. These results suggest that our proposal may act as a controllable thermospin generator.     

Distributionally robust inference for extreme Value-at-Risk

Under general multivariate regular variation conditions, the extreme Value-at-Risk of a portfolio can be expressed as an integral of a known kernel with respect to a generally unknown spectral measure supported on the unit simplex. The estimation of the spectral measure is challenging in practice and virtually impossible in high dimensions. This motivates the problem studied in this work, which is to find universal lower and upper bounds of the extreme Value-at-Risk under practically estimable constraints. That is, we study the infimum and supremum of the extreme Value-at-Risk functional, over the infinite dimensional space of all possible spectral measures that meet a finite set of constraints. We focus on extremal coefficient constraints, which are popular and easy to interpret in practice. Our contributions are twofold. First, we show that optimization problems over an infinite dimensional space of spectral measures are in fact dual problems to linear semi-infinite programs (LSIPs) – linear optimization problems in Euclidean space with an uncountable set of linear constraints. This allows us to prove that the optimal solutions are in fact attained by discrete spectral measures supported on finitely many atoms. Second, in the case of balanced portfolia, we establish further structural results for the lower bounds as well as closed form solutions for both the lower- and upper-bounds of extreme Value-at-Risk in the special case of a single extremal coefficient constraint. The solutions unveil important connections to the Tawn–Molchanov max-stable models. The results are illustrated with two applications: a real data example and closed-form formulae in a market plus sectors framework.     

Response surface optimisation applied to a headspace-solid phase microextraction-gas chromatography-mass spectrometry method for the analysis of volatile organic compounds in water matrices

A new method for the simultaneous determination of 12 volatile organic compounds (trans-1,2-dichloroethene, 1,1,1-trichloroethane, benzene, 1,2-dichloroethane, trichloroethene, toluene, 1,1,2-trichloroethane, tetrachloroethene, ethylbenzene, m-, p-, o-xylene) in water samples by headspace solid phase microextraction (HS–SPME)–gas chromatography mass spectrometry (GC–MS) was described, using a 100?µm PDMS (polydimethylsiloxane) coated fibre. The response surface methodology was used to optimise the effect of the extraction time and temperature, as well as the influence of the salt addition in the extraction process. Optimal conditions were extraction time and temperature of 30?min and ?20°C, respectively, and NaCl concentration of 4?mol?L^?1. The detection limits were in the range of 1.1?×?10^?3–2.3?µg?L^?1 for the 12 volatile organic compounds (VOCs). Global uncertainties were in the range of 4–68%, when concentrations decrease from 250?µg?L^?1 down to the limits of quantification. The method proved adequate to detect VOCs in six river samples.